Let $X$ be a quasi-projective variety, $Y$ a projective variety, and $f:X \rightarrow Y$ an open immersion. If $\mathcal{F}$ is a locally free coherent sheaf, what can be said about $f_\ast \mathcal{F}$? Is it coherent? Is it torsion free? Is it reflexive?
[Math] Is the pushforward of a locally free sheaf by an open immersion coherent
ag.algebraic-geometry
Best Answer
About your new question:
Let $Y$ be a projective variety and let $X\subset Y$ be an open subset with complement the closed subset $S:=Y\setminus X$. Call $f:X\hookrightarrow Y$ the inclusion.
Let $\mathcal F$ be an algebraic coherent sheaf without torsion on $X$.
Theorem (Serre-Grothendieck) Suppose that $Y$ is normal and that $S$ has codimension $\geq 2$. Then the sheaf $f_\ast \mathcal F$ is coherent.
Serre, Prolongement de faisceaux analytiques cohérents, Ann.Inst.Fourier 16 (1966), 363-374